264 CHAPTER 8 GEOMETRY FOR AMERICANS
Circles and Angles
Circles are the most important entities in Euclidean geometry. You cannot learn too
much about them. We shall refer to the figure below to review the most important
terms.rWe denote the circle by r. Its center is 0, so L.COB is called a central angle. Line
segment BC is called a chord, while BC refers to the arc from B to C. As usual, we
may informally write "arc BC" in its place.^3 The measure of arc BC is defined to be
equal to the measure of the central angle that subtends it, i.e.mBC= mL.COB.
Because A lies on the circle, we call L.CAB an inscribed angle; we also say that this
angle subtends arc BC at the circle.
Line DB intersects the circle in exactly one point, namely B, so this line is called
a tangent.Fact 8.2.13 Relationship Between Chords and Radii.^4 Consider the following three
statements about a line, a circle, and a chord of the circle:1. The line passes through the center of the circle.
(^2). The line passes through the midpoint of the chord.
3. The line is perpendicular to the chord.
If any two of these statements are true, then all three are true.
Fact 8.2. 14 A tangent line to a circle is perpendicular to the radius at the point of
tangency; conversely, the perpendicular to a tangent line at the point of tangency will
pass through the center of the circle.8.2.15 Let A be a point outside a circle. Draw lines AX, AY tangent to the circle at X
and Y. Prove that AX = AY.
(^3) Notice that there is an ambiguity about arcs. Does arc Be mean the path along the circumference going
clockwise or counterclockwise? If it is the former, the central angle will be more than 1800 • We will use the
convention that arcs are read counterclockwise.
(^4) We are indebted to Andy Liu, who used this clever "three for two sale" formulation in [28].